When I started as a graduate student in mathematics lo these many moons ago, there were maybe four open questions which were then considered to be the pinnacle of mathematics, analogous to scaling Everest or running a sub-four-minute mile. All of them had been unsolved for decades or even centuries, despite the best efforts of the strongest minds. Since then two or the four (the Bieberbach Conjuecture and the Shimura-Taniyama Conjecture (Fermat's Last Theorem)) have been solved by stunning and surprising approaches. Of the two remaining, the Poincare Conjecture and the Riemann Hypothesis, it would appear that the Poincare Conjecure has fallen. It is odd that we are living in the abolute golden age of mathematics and yet almost no one knows it or even has the merest awareness of what that means.

This ignorance occurs for good, rather than nefarious, reasons. Mathematics has basically become too abstract and too sophisticated to be able to be explained any more. It takes years of arduous study just to learn what the terms mean. Yet it has been shown in the last quarter century or so that even the most abstruse and recondite piece of pure mathematics is likely to have important consequences or applications in the physical world, so it seems to me incumbent upon us as citizens of a democracy to try to understand something about the froth of activity going on surreptitiously around us. Continued....Allow me then to take a crack at explaining the Poincare Conjecture: the only 3-dimensional object that has the shape of a sphere is a sphere. That's deceptively simple, because I'm not talking about the ordinary sphere (called the 2-sphere in mathematics because if you live on one, say the surface of the Earth, your movements are constrained to only 2 dimensions), but rather its analog in 4-space (the set of all points in Euclidean 4-space whose distance is one from the origin is the unit 3-sphere). And by having the "shape" of the sphere I mean that its shape as measured by a very important mathematical yardstick called the "fundamental group" as well as by the "homology groups" is that of a sphere. What this Conjecture does is validate our ordinary intuition in the strongest way. That the Conjecture is true there has been very little doubt for a long time; that we found ourselves unable to prove it has been a huge embarassment to the species. The Conjuecture means that this yardstick really captures the topological essence of a sphere, which is really good news because the fundamental group is far simpler than a sphere and easier to work with.

The proof itself has an interesting human interest story behind it. Like many such intellectual feats (think chessmasters) it emanates not from the Anglo-Saxon world but from Russia, with its Orthodox, Greek-based culture. The author, Perelman, of this alleged proof wrote a couple of short papers claiming to have a proof. He wasn't quite believed because the papers were too short and surprising to be convincing. He went on a whirlwind tour of the US to make his case; American mathematicians have been working hard for three years now to fill in the numerous gaps in Perelman's ideas, and Perelman has apparently completely disappeared. There's a large prize of a million dollars to be had should he reappear, together with (probably) a Fields Medal, the mathematical equivalent of the Nobel Prize. “He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”

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You might mention that, speaking broadly, every surface has an associated fundamental group. A surface is a geometric thing, the fundamental group is an algebraic thing. What one would like is that different surfaces have different fundamental groups so that the groups can be used to uniquely identify them. Unfortunately, that is not generally the case. But for the very simplest group, which contains only the number 1 (1 x 1 = 1)*, it is now known that the only bounded surface with this group is the sphere. And this holds in all dimensions greater than one.

*groups have only one operation, shown here as multiplication. Those algebraic things that also have addition are, again broadly speaking, called rings. The whole numbers, for instance. There is also a way to associate rings with surfaces, the result is called the cohomology ring of the surface. The whole subject of tagging surfaces with algebraic objects is called algebraic topology.

“Didn't I read somewhere that mathematics is the language most likely to be spoken by alien forms of life?”

You are probably referring to the Jodie Foster movie, Contact. Math is a hard science activity and mostly useless in human communications. We humans add and subtract words on a constant basis. Our use of irony drives computer programs crazy. Furthermore, homo sapiens also “speak” via their facial expressions and overall body language. This is why computer language interpretation programs leave something to be desired and can never be perfect.

Geometry is true. It just isn't the whole truth. The same with f = ma.

Mathematics explains what is and also presents us with the is we don't yet know or an is that is parallel because of conditions we're not aware of.

Even though it takes great brain power and dedication to understand mathematics, the truths do trickle down to us who live in less rarified atmospheres.

I remember in highschool solving equations that had two solutions. The teacher said 'use this one. throw that one out'. Why? I asked. What makes one solution real and one not?

He told me they're both real but based on different outside conditions. In this case the equation had to do with the bounce (or not) of a cannon ball. The solution we threw out solved for an earth with no surface where the cannonball would pass right through.

If we could not conceive of no surface to the earth that would stop a cannonball, we would look on that solution as weird, crazy, strange, unexplainable.

Until somebody had an aha moment and laid it out.

What's so heartening about the subject here is the opposite in a way, and is that the proof seems to be in about something that we intuitively understand.

Not that intuition is always reliable, but heck I'd rather intuition be proved correct than face the jolt of the opposite.